The arson attack at the park in Craven Road took place over the weekend, causing extensive damage to a wooden climbing frame.
Planks of wood and a petrol can were found at the scene. The council, which has reported the attack to the police, has now fenced off the park's play equipment until repairs can be carried out.
In recent weeks the council has dealt with an increasing number of arson attacks, vandalism and anti-social behaviour at the borough's parks and open spaces.
Vandals caused thousands of pounds of damage to play equipment at Whitehall Recreation Ground earlier this month, while play equipment at Newbold's Centenary Park, Hillmorton's GEC Recreation Ground and the neighbourhood play areas at Cawston and Coton Park has also been repeatedly targeted.
Arsonists set fire to a wheelie bin on the multi-use games area at Jubilee Street Recreation Ground, damaging the tarmac, while broken glass has become a significant issue at parks and play areas across the borough.
And the council's parks and grounds team has repeatedly faced clearing human faeces from Caldecott Park in the town centre, together with discarded toilet roll.
Cllr Howard Roberts, Rugby Borough Council portfolio holder for environment and public realm, said the spate of vandalism and anti-social behaviour was putting extra pressure on resources at a time when the council was playing a vital role in dealing with the coronavirus pandemic.
"Since the start of lockdown our parks and open spaces have proved a lifeline for many of our residents, so it's frustrating the actions of a mindless few deprive the majority of a safe and welcoming environment," Cllr Roberts added.
"Throughout our response to the pandemic, we've urged our residents to do what's 'Right for Rugby' and it's been heartening to see our communities come together during these difficult times.
"It makes such behaviour from a selfish minority hard to understand and even harder to tolerate, and we're determined to stamp it out and catch the culprits."
Anyone with information about the arson attack at Millennium Green should call Warwickshire Police on 101, quoting crime number 23/40338/20, or the confidential Crimestoppers hotline on 0800 555 111.